Theorem sepnsepolem2 49586

Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 49587. Proof could be shortened by 1 step using ssdisjdr 49472. (Contributed by Zhi Wang, 1-Sep-2024.)

Hypothesis

Ref	Expression
sepnsepolem2.1	⊢ (𝜑 → 𝐽 ∈ Top)

Assertion

Ref	Expression
sepnsepolem2	⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Distinct variable groups:   𝑦,𝐷   𝑦,𝐽   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐷(𝑥)   𝐽(𝑥)

Proof of Theorem sepnsepolem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sepnsepolem2.1	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
2		id 23	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
3		sslin 4203	. . . . 5 ⊢ (𝑧 ⊆ 𝑦 → (𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦))
4		sseq0 4367	. . . . . 6 ⊢ (((𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑧) = ∅)
5	4	ex 417	. . . . 5 ⊢ ((𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
6	3, 5	syl 18	. . . 4 ⊢ (𝑧 ⊆ 𝑦 → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
7	6	adantl 486	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
8		ineq2 4175	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝑧))
9	8	eqeq1d 2771	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
10	9	adantl 486	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 = 𝑧) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
11	2, 7, 10	opnneieqv 49574	. 2 ⊢ (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
12	1, 11	syl 18	1 ⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ‘cfv 6537  Topctop 23019  neicnei 23223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23020  df-nei 23224

This theorem is referenced by:  sepnsepo  49587